Question

A naughty student breaks the pencil in such a way that the ratio of two broken parts is same as that of the original length of the pencil to one of the larger part of the pencil, The ratio of the other part to the original length of pencil is:
A.$1:2\sqrt{5}$
B.$2:(3+\sqrt{5})$
C.$2:\sqrt{5}$
D.Can’t be determined

Answer 1

Hint: Here, assume the length of the larger part be $x$ and that of the smaller part be $y$. Then apply the condition according to the problem and simplify it.

Complete step-by-step answer:
Let us assume that the length of the larger part be $x$ and that of the smaller part be $y$.
Now according to condition, their ratio is
$\dfrac{x}{y}=\dfrac{x+y}{x}$
Now simplifying we get,
$\Rightarrow$ ${{x}^{2}}=xy+{{y}^{2}}$
$\Rightarrow$ ${{x}^{2}}-{{y}^{2}}=xy$ …………. (1)
Now let us substitute $y=1$ in equation (1) we get,
$\Rightarrow$ ${{x}^{2}}-{{1}^{2}}=x$
Now rearranging the equation we get,
$\Rightarrow$ ${{x}^{2}}-x-1=0$
Now comparing ${{x}^{2}}-x-1=0$ with $a{{x}^{2}}+bx+c=0$.
So here $a=1,b=-1$ and $c=-1$.
Now using $x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$
So substituting all the values we get,
$\Rightarrow$ $x=\dfrac{-(-1)\pm \sqrt{{{(-1)}^{2}}-4(1)(-1)}}{2(1)}$
Now simplifying we get,
$\Rightarrow$ $x=\dfrac{1\pm \sqrt{5}}{2}$
Here we have got two values but negative values cannot be consider, so the ratio becomes,
$\Rightarrow$ $\dfrac{x}{y}=\dfrac{\dfrac{1+\sqrt{5}}{2}}{1}$
$\Rightarrow$ $\dfrac{x}{y}=\dfrac{1+\sqrt{5}}{2}$
Now taking componendo we get,
$\Rightarrow$ $\dfrac{x+y}{y}=\dfrac{1+\sqrt{5}+2}{2}$
$\Rightarrow$ $\dfrac{x+y}{y}=\dfrac{3+\sqrt{5}}{2}$
Now taking reciprocal,
$\Rightarrow$ $\dfrac{y}{x+y}=\dfrac{2}{3+\sqrt{5}}$
The ratio of the other part to the original length of pencil is $\dfrac{2}{3+\sqrt{5}}$.
The correct answer we get is option (B).

Additional information:
Ratio and Proportion are explained majorly based on fractions. When a fraction is represented in the form of a:b, then it is a ratio whereas a proportion states that two ratios are equal. Here, a and b are any two integers. Proportion is an equation which defines that the two given ratios are equivalent to each other. The two numbers in a ratio can only be compared when they have the same unit. We make use of ratios to compare two things. Ratio and proportions are said to be faces of the same coin. When two ratios are equal in value, then they are said to be in proportion. The ratio should exist between the quantities of the same kind.

Note: The ratio is used to compare the size of two things with the same unit. The proportion is used to express the relation of two ratios. So if we have $\dfrac{a}{b}=\dfrac{c}{d}$ so componendo will be $\dfrac{a+b}{b}=\dfrac{c+d}{d}$.